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Example 4 from Algebra and Trigonometry, 4.2 Modeling with Linear Functions
Building a System of Linear Models to Choose a Truck Rental Company
Jamal is choosing between two truck-rental companies. The first, Keep on Trucking, Inc., charges an up-front fee of $20, then 59 cents a mile. The second, Move It Your Way, charges an up-front fee of $16, then 63 cents a mile4. When will Keep on Trucking, Inc. be the better choice for Jamal?
4
Rates retrieved Aug 2, 2010 from http://www.budgettruck.com and http://www.uhaul.com/Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.
Solution
The two important quantities in this problem are the cost and the number of miles driven. Because we have two companies to consider, we will define two functions in Table 1.
Input distance driven in miles Outputs cost, in dollars, for renting from Keep on Trucking
cost, in dollars, for renting from Move It Your Way
Initial Value Up-front fee: and Rate of Change /mile and /mile Table 1A linear function is of the form Using the rates of change and initial charges, we can write the equations
Using these equations, we can determine when Keep on Trucking, Inc., will be the better choice. Because all we have to make that decision from is the costs, we are looking for when Move It Your Way, will cost less, or when The solution pathway will lead us to find the equations for the two functions, find the intersection, and then see where the function is smaller.
These graphs are sketched in Figure 6, with in blue.
Figure 6To find the intersection, we set the equations equal and solve:
This tells us that the cost from the two companies will be the same if 100 miles are driven. Either by looking at the graph, or noting that is growing at a slower rate, we can conclude that Keep on Trucking, Inc. will be the cheaper price when more than 100 miles are driven, that is .
How did it go?